Import Question JSON

Current Question (ID: 20005)

Question:

$\text{Two waves executing simple harmonic motion travelling in the same direction with the same amplitude and frequency are superimposed.}$ $\text{The resultant amplitude is equal to the } \sqrt{3} \text{ times of amplitude of individual motions.}$ $\text{The phase difference between the two motions is:}$

Options:

1. $30^\circ$
2. $45^\circ$
3. $60^\circ$
4. $90^\circ$

Solution:

$\text{When two waves of the same amplitude } A \text{ and frequency are superimposed, the resultant amplitude } R \text{ is given by:}$ $R = 2A \cos\left(\frac{\phi}{2}\right)$ $\text{Given } R = \sqrt{3}A, \text{ we have:}$ $\sqrt{3}A = 2A \cos\left(\frac{\phi}{2}\right)$ $\cos\left(\frac{\phi}{2}\right) = \frac{\sqrt{3}}{2}$ $\frac{\phi}{2} = 30^\circ$ $\phi = 60^\circ$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}